A1g

A1g is a label used in group theory and molecular symmetry to denote a specific irreducible representation (irrep) of a point group. It appears in several centrosymmetric point groups, such as Oh, D4h, D2h, and C2h, where it helps classify electronic states, molecular vibrations, and other properties according to symmetry.

The A1g representation is typically one-dimensional, meaning it is nondegenerate and transforms as a single basis function under the group’s symmetry operations. The subscript g stands for gerade, indicating even parity under inversion (the function remains unchanged when inverted through the center of symmetry). The letter A signals that the irrep is nondegenerate, while the number 1 distinguishes it from other one-dimensional representations within the same group.

In practice, A1g is often associated with totally symmetric behavior. Basis functions that transform according to A1g are invariant under all symmetry operations of the group; common examples include functions like z^2 and x^2 + y^2 in appropriate groups (for example, D4h) and the constant function 1 in many groups. This makes A1g frequently relevant for describing Raman-active modes and other properties that peak in symmetry-preserving transformations.

Use of A1g extends to spectroscopy and crystallography, where it helps assign vibrational modes and electronic states, predict selection rules, and interpret spectra. The concept is part of the broader Mulliken notation system for labeling irreducible representations in point groups. See also: point group, Mulliken notation, Raman spectroscopy.